Classification of diffeomorphisms by association of differentials with Lie groups

Question

Suppose we are given an oriented Riemannian manifold $S \subset \mathbb{R}^3$ (which I'll refer to as a surface) and a diffeomorphism on $S$, $\Psi: S \rightarrow S$ where $d\Psi\vert_{\bf q}:T_{{\bf q}}S \rightarrow T_{\Psi({\bf q})}S$ is the differential of $\Psi$ evaluated at ${\bf q} \in S$. For illustrative purposes, we'll consider local neighborhoods $N_{\bf p}, N_{\Psi({\bf p})} \subset S$ about an arbitrary point ${\bf p} \in S$ and $\Psi({\bf p}) \in S$.

If $\Psi$ is a local isometry, then $\forall {\bf q} \in N_{\bf p}$, $d\Psi\vert_{\bf q}$ can be associated with transformation in $\textrm{SO}(2)$ as $d\Psi$ preserves the inner product $$\langle {\bf v}_1, \ {\bf v}_2\rangle = \langle \ [d\Psi\vert_{\bf q}] {\bf v}_1, \ [d\Psi\vert_{\bf q}] {\bf v}_2 \rangle, $$ for all ${\bf v}_1, {\bf v}_2 \in T_{\bf q}S$.

Similarly, if $\Psi$ is locally conformal, there exists a differentiable function $\lambda^2:N_{\bf p}\rightarrow \mathbb{R}_{>0}$ such that for $\forall {\bf q} \in N_{\bf p}$, $$ \lambda^2({\bf q}) \langle {\bf v}_1, \ {\bf v}_2\rangle = \langle \ [d\Psi\vert_{\bf q}] {\bf v}_1, \ [d\Psi\vert_{\bf q}]{\bf v}_2 \rangle,$$for all ${\bf v}_1, {\bf v}_2 \in T_{\bf q}S$. It follows that for each ${\bf q} \in N_{\bf p}$, $d\Psi\vert_{\bf q}$ can be associated with an element of the Lie group $$\left\{ \alpha R \in \mathbb{R}^{2 \times 2} \ \mid \ R \in \textrm{SO}(2), \ \alpha \in \mathbb{R}_{>0} \right\}.$$

My question is as follows:

The above examples suggest that at least some types of diffeomorphisms on surfaces can be classified by associating the differential with a planar Lie group.

It seems that a natural next step would be to a define classes of diffeomorphisms whose differentials can be associated with $\textrm{SL}(2, \mathbb{R})$ and $\textrm{GL}(2, \mathbb{R})$, with the former possibly preserving something like local surface areas and the latter a notion of handedness.

I've looked around a bit but haven't yet been able to find a comprehensive treatment of diffeomorphisms that considers more "complex" types of transformations than isometries and conformal mappings, let alone anything that approaches the topic from more of a matrix Lie group perspective as I described above.

I'm hoping that someone might be able give me some information about any classes of mappings possibly associated with higher dimensional planar Lie groups (i.e. $\textrm{SL}(2, \mathbb{R}), \ \textrm{GL}(2, \mathbb{R})$ ). However, my knowledge of Riemannian/conformal geometry could charitably be described as limited, so it is likely that I'm unaware of well-known types of diffeomorphisms that fit the bill. In any case, pointing me towards a few resources that provide an in-depth treatment of more general classes of diffeomorphisms would be greatly appreciated.

Answer 1

Much of this question is linear-algebraic in flavor:

For any point $p$ on a Riemannian manifold $(M, g)$ of dimension $n$, the tangent space $(T_p M, g_p)$ is an inner product space. Certain bases $(E_a)$ of $T_p M$ are well-adapted to the inner product, namely its orthonormal bases, those that satisfy $$g_p(E_a, E_a) = 1 \qquad \textrm{and} \qquad g_p(E_a, E_b) = 0$$ for all $a, b$, $a \neq b$. At least when $n > 1$, there are many choices of orthonormal basis of $(T_p M, g_p)$, and given any isometry $\phi : T_p M \to T_p M$ of $g_p$, we have $$g_p(E_a, E_b) = (\phi^* g_p)(E_a, E_b) = g_p(\phi(E_a), \phi(E_b)) ;$$ in particuar, if $(E_a)$ is an orthonormal basis, so is $(\phi(E_a))$. Thus, the group $O(g_p) \cong O(n, \Bbb R)$ acts transitively (and, in fact, freely) on the space (which below we'll denote $\mathcal F^O_p$) of orthonormal bases.

Back at the level of the Riemannian manifold $(M, g)$, unwinding definitions shows that the following are equivalent:

• a diffeomorphism $\Phi : M \to M$ is an isometry;
• for every $p \in M$ the differential $T_p \Phi$ is an isometry $(T_p M, g_p) \to (T_{\Phi(p)} M, g_{\Phi(p)})$ of inner product spaces;
• for every $p \in M$ and any (equivalently, every) orthonormal basis $(E_a)$ of $T_p M$, $(T_p \Phi \cdot E_a)$ is an orthonormal basis of $T_{\Phi(p)} M$.

Put informally, a diffeomorphism is an isometry if it takes orthonormal bases to orthonormal bases.

This perspective suggests repackaging these ideas as follows:

For any smooth manifold $M$, the (tangent) frame bundle is the fiber bundle $\mathcal{F} \to M$ whose fiber $\mathcal F_p$ over $p$ consists of the bases of $T_p M$. The defining action of $GL(T_p M)$ on $T_p M$ takes bases to bases, so the induced action on the space $\mathcal F_p$ of bases realize $\mathcal F$ as a principal $GL(n)$-bundle over $M$. By definition, we may identify sections of this bundle with frames on $M$.

Likewise, for any Riemannian manifold $(M, g)$, the canonical orthonormal frame bundle is the bundle $\mathcal F^O \to M$ whose fiber $\mathcal F^O_p$ over $p$ consists of the orthonormal bases of $(T_p M, g_p)$, and by construction at each $p \in M$ the action of $GL(T_p M)$ restricts to the action of $O(g_p)$ described at the beginning of the answer. We may identify sections of $\mathcal F^O \to M$ with orthonormal frames.

In this language, any diffeomorphism $\Phi: M \to M$ induces a bundle isomorphism $\hat\Phi: \mathcal F \to \mathcal F$, and if $M$ is equipped with a Riemannian metric $g$, it is an isometry iff it maps $\mathcal F^O$ to itself. We call $g$---or, just as well, the frame bundle $\mathcal F^O$---an $O(n)$-structure.

Conversely, if we had started out with the $O(n)$-structure $\mathcal F^O$, we could have reconstructed the Riemannian metric $g$, and, as you suggest, we can ask what geometries can be realized as $G$-structures for other Lie subgroups $G \leq GL(n, \Bbb R)$, and for each what is the space of compatible bases/frames. For example:

• The subgroup $GL_+(n, \Bbb R)$ of linear transformations of positive determinant corresponds to an orientation; the compatible bases are the positively oriented ones.
• The subgroup $SL(n, \Bbb R)$ corresponds to a a volume form on $M$, i.e. a nonvanishing section $\Omega$ of $\bigwedge^n T^*M$; the compatible bases $(E_a)$ are those that span a parallelepiped of unit volume, i.e., for which $\Omega(E_1, \ldots, E_n) = 1$.
• The subgroup $CO(n, \Bbb R)$ corresponds to a conformal structure $[g]$ on $M$; the compatible bases at $p$ are those that are an orthonormal for some metric $g_p$ in $[g_p]$.
• The subgroup $GL(\frac{n}{2}, \Bbb C)$ corresponds to an almost complex structure, that is an endomorphism field $J : TM \to TM$ satisfying $J^2 = -1$ (essentially, this is an identification of each tangent space $T_p M$ with a complex vector space of dimension $\frac{n}{2}$). One natural choice for compatible bases are those of the form $(E_1, JE_1, \ldots, E_{n / 2}, J E_{n / 2})$ for some $(E_1, \ldots, E_{n / 2})$.

In all of the above examples, the definition of the geometric structure is essentially linear-algebraic, in that it can be characterized separately at each point (requiring only additionally that the structure vary smoothly from point to point). But many geometric structures are defined in part by differential conditions (often we call these integrability or nonintegrability conditions, depending on their character). For example, an almost complex structure $(M, J)$ defines a complex structure (i.e., a compatible maximal atlas of holomorphic charts) iff a certain tensor $N_J : \bigwedge^2 TM \to TM$---which in particular depends on the derivative of $J$---vanishes. In many cases, $G$-structures come equipped with a canonical connection (this is case for $G = O(n)$, i.e., for Riemannian manifolds, in which case the canonical connection is essentially the Levi-Civita connection), which can be used to study the differential behavior of the structure.